HomeHome Metamath Proof Explorer
Theorem List (p. 162 of 311)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-31058)
 

Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempsr1bas2 16101 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  S )   &    |-  O  =  ( 1o mPwSer  R )   =>    |-  B  =  ( Base `  O )
 
Theorempsr1bas 16102 The base set of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  S  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  S )   &    |-  K  =  (
 Base `  R )   =>    |-  B  =  ( K  ^m  ( NN0  ^m 
 1o ) )
 
Theoremvr1val 16103 The value of the generator of the power series algebra (the  X in  R [ [ X ] ]). Since all univariate polynomial rings over a fixed base ring  R are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and  1o  =  { (/) } is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
 |-  X  =  (var1 `  R )   =>    |-  X  =  ( ( 1o mVar  R ) `  (/) )
 
Theoremvr1cl2 16104 The variable  X is a member of the power series algebra  R [ [ X ] ]. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  X  =  (var1 `  R )   &    |-  S  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  S )   =>    |-  ( R  e.  Ring  ->  X  e.  B )
 
Theoremply1val 16105 The value of the set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  S  =  (PwSer1 `  R )   =>    |-  P  =  ( Ss  (
 Base `  ( 1o mPoly  R ) ) )
 
Theoremply1bas 16106 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  S  =  (PwSer1 `  R )   &    |-  U  =  ( Base `  P )   =>    |-  U  =  ( Base `  ( 1o mPoly  R )
 )
 
Theoremply1lss 16107 Univariate polynomials form a linear subspace of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  S  =  (PwSer1 `  R )   &    |-  U  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  U  e.  ( LSubSp `  S ) )
 
Theoremply1subrg 16108 Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  S  =  (PwSer1 `  R )   &    |-  U  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  U  e.  (SubRing `  S ) )
 
Theoremply1crng 16109 The ring of univariate polynomials is a commutative ring. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  CRing  ->  P  e.  CRing )
 
Theoremply1assa 16110 The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  CRing  ->  P  e. AssAlg )
 
Theorempsr1rclOLD 16111 Obsolete version of elbasfv 13065 as of 5-Apr-2016. Reverse closure for ring existence from the univariate power series base set. (Contributed by Stefan O'Rear, 25-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  P  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  R  e.  _V )
 
Theorempsr1bascl 16112 A univariate power series is a multivariate power series on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  P  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  F  e.  ( Base `  ( 1o mPwSer  R )
 ) )
 
Theorempsr1basf 16113 Univariate power series base set elements are functions. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  P  =  (PwSer1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( F  e.  B  ->  F : (
 NN0  ^m  1o ) --> K )
 
Theoremply1rclOLD 16114 Obsolete version of elbasfv 13065 as of 5-Apr-2016. Reverse closure for ring existence from the univariate polynomial base set. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  R  e.  _V )
 
Theoremply1basf 16115 Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( F  e.  B  ->  F : (
 NN0  ^m  1o ) --> K )
 
Theoremply1bascl 16116 A univariate polynomial is a univariate power series. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  F  e.  ( Base `  (PwSer1 `  R ) ) )
 
Theoremply1bascl2 16117 A univariate polynomial is a multivariate polynomial on one index. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( F  e.  B  ->  F  e.  ( Base `  ( 1o mPoly  R )
 ) )
 
Theoremcoe1fval 16118* Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   =>    |-  ( F  e.  V  ->  A  =  ( n  e.  NN0  |->  ( F `
  ( 1o  X.  { n } ) ) ) )
 
Theoremcoe1fv 16119 Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   =>    |-  ( ( F  e.  V  /\  N  e.  NN0 )  ->  ( A `  N )  =  ( F `  ( 1o  X.  { N } ) ) )
 
Theoremfvcoe1 16120 Value of a multivariate coefficient in terms of the coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   =>    |-  ( ( F  e.  V  /\  X  e.  ( NN0  ^m  1o ) ) 
 ->  ( F `  X )  =  ( A `  ( X `  (/) ) ) )
 
Theoremcoe1fval3 16121* Univariate power series coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (PwSer1 `  R )   &    |-  G  =  ( y  e.  NN0  |->  ( 1o 
 X.  { y } )
 )   =>    |-  ( F  e.  B  ->  A  =  ( F  o.  G ) )
 
Theoremcoe1f2 16122 Functionality of univariate power series coefficient vectors. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (PwSer1 `  R )   &    |-  K  =  (
 Base `  R )   =>    |-  ( F  e.  B  ->  A : NN0 --> K )
 
Theoremcoe1fval2 16123* Univariate polynomial coeffecient vectors expressed as a function composition. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (Poly1 `  R )   &    |-  G  =  ( y  e.  NN0  |->  ( 1o 
 X.  { y } )
 )   =>    |-  ( F  e.  B  ->  A  =  ( F  o.  G ) )
 
Theoremcoe1f 16124 Functionality of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (Poly1 `  R )   &    |-  K  =  (
 Base `  R )   =>    |-  ( F  e.  B  ->  A : NN0 --> K )
 
Theoremcoe1sfi 16125 Finite support of univariate polynomial coefficient vectors. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  A  =  (coe1 `  F )   &    |-  B  =  ( Base `  P )   &    |-  P  =  (Poly1 `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( F  e.  B  ->  ( `' A "
 ( _V  \  {  .0.  } ) )  e. 
 Fin )
 
Theoremvr1cl 16126 The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  X  =  (var1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  X  e.  B )
 
Theoremopsr0 16127 Zero in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  ( 0g `  S )  =  ( 0g `  O ) )
 
Theoremopsr1 16128 One in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  ( 1r `  S )  =  ( 1r `  O ) )
 
Theoremmplplusg 16129 Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  .+  =  ( +g  `  Y )   =>    |-  .+  =  ( +g  `  S )
 
TheoremmplvscafvalOLD 16130 Obsolete version of mplvsca2 16022 as of 5-Apr-2016. Value of scalar multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  Y  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .s `  Y )   =>    |-  .x.  =  ( .s `  S )
 
Theoremmplmulr 16131 Value of multiplication in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  Y  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .r `  Y )   =>    |-  .x.  =  ( .r `  S )
 
Theorempsr1plusg 16132 Value of addition in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  Y  =  (PwSer1 `  R )   &    |-  S  =  ( 1o mPwSer  R )   &    |-  .+  =  ( +g  `  Y )   =>    |-  .+  =  ( +g  `  S )
 
Theorempsr1vsca 16133 Value of scalar multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  Y  =  (PwSer1 `  R )   &    |-  S  =  ( 1o mPwSer  R )   &    |-  .x.  =  ( .s `  Y )   =>    |-  .x.  =  ( .s `  S )
 
Theorempsr1mulr 16134 Value of multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  Y  =  (PwSer1 `  R )   &    |-  S  =  ( 1o mPwSer  R )   &    |-  .x.  =  ( .r `  Y )   =>    |-  .x.  =  ( .r `  S )
 
Theoremply1plusg 16135 Value of addition in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  S  =  ( 1o mPoly  R )   &    |-  .+  =  ( +g  `  Y )   =>    |-  .+  =  ( +g  `  S )
 
Theoremply1vsca 16136 Value of scalar multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  S  =  ( 1o mPoly  R )   &    |-  .x.  =  ( .s `  Y )   =>    |-  .x.  =  ( .s `  S )
 
Theoremply1mulr 16137 Value of multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  S  =  ( 1o mPoly  R )   &    |-  .x.  =  ( .r `  Y )   =>    |-  .x.  =  ( .r `  S )
 
Theoremressply1bas2 16138 The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  W  =  (PwSer1 `  H )   &    |-  C  =  ( Base `  W )   &    |-  K  =  ( Base `  S )   =>    |-  ( ph  ->  B  =  ( C  i^i  K ) )
 
Theoremressply1bas 16139 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  P  =  ( Ss  B )   =>    |-  ( ph  ->  B  =  ( Base `  P )
 )
 
Theoremressply1add 16140 A restricted polynomial algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
 
Theoremressply1mul 16141 A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( .r `  U ) Y )  =  ( X ( .r `  P ) Y ) )
 
Theoremressply1vsca 16142 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  P  =  ( Ss  B )   =>    |-  ( ( ph  /\  ( X  e.  T  /\  Y  e.  B )
 )  ->  ( X ( .s `  U ) Y )  =  ( X ( .s `  P ) Y ) )
 
Theoremsubrgply1 16143 A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  (Poly1 `  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( T  e.  (SubRing `  R )  ->  B  e.  (SubRing `  S ) )
 
Theorempsrbaspropd 16144 Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  ( Base `  R )  =  ( Base `  S )
 )   =>    |-  ( ph  ->  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  S ) ) )
 
Theorempsrplusgpropd 16145* Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( I mPwSer  S ) ) )
 
Theoremmplbaspropd 16146* Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  S ) ) )
 
Theoremstrov2rcl 16147 Reverse closure for polynomial-resembling things. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  S  =  ( I F R )   &    |-  B  =  ( Base `  S )   &    |-  Rel  dom 
 F   =>    |-  ( X  e.  B  ->  I  e.  _V )
 
Theorempsropprmul 16148 Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  Y  =  ( I mPwSer  R )   &    |-  S  =  (oppr `  R )   &    |-  Z  =  ( I mPwSer  S )   &    |-  .x.  =  ( .r `  Y )   &    |-  .xb  =  ( .r `  Z )   &    |-  B  =  ( Base `  Y )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  =  ( G  .x.  F ) )
 
Theoremply1opprmul 16149 Reversing multiplication in a ring reverses multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  S  =  (oppr `  R )   &    |-  Z  =  (Poly1 `  S )   &    |- 
 .x.  =  ( .r `  Y )   &    |-  .xb  =  ( .r `  Z )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .xb  G )  =  ( G  .x.  F ) )
 
Theorem00ply1bas 16150 Lemma for ply1basfvi 16151 and deg1fvi 19303. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  (/)  =  ( Base `  (Poly1 `  (/) ) )
 
Theoremply1basfvi 16151 Protection compatibility of the univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  (  _I  `  R ) ) )
 
Theoremply1plusgfvi 16152 Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  (  _I  `  R )
 ) )
 
Theoremply1baspropd 16153* Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  S ) ) )
 
Theoremply1plusgpropd 16154* Property deduction for univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  S )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  R )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( +g  `  (Poly1 `  R ) )  =  ( +g  `  (Poly1 `  S ) ) )
 
Theoremopsrrng 16155 Ordered power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  e.  Ring )
 
Theoremopsrlmod 16156 Ordered power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  O  =  ( ( I ordPwSer  R ) `  T )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  T 
 C_  ( I  X.  I ) )   =>    |-  ( ph  ->  O  e.  LMod )
 
Theorempsr1rng 16157 Univariate power series form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  S  =  (PwSer1 `  R )   =>    |-  ( R  e.  Ring  ->  S  e.  Ring )
 
Theoremply1rng 16158 Univariate polynomials form a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  Ring )
 
Theorempsr1lmod 16159 Univariate power series form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  LMod )
 
Theorempsr1sca 16160 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 4-Jul-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  ( R  e.  V  ->  R  =  (Scalar `  P ) )
 
Theorempsr1sca2 16161 Scalars of a univariate power series ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  (PwSer1 `  R )   =>    |-  (  _I  `  R )  =  (Scalar `  P )
 
Theoremply1lmod 16162 Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  Ring  ->  P  e.  LMod )
 
Theoremply1sca 16163 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e.  V  ->  R  =  (Scalar `  P ) )
 
Theoremply1sca2 16164 Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  (  _I  `  R )  =  (Scalar `  P )
 
Theoremply1mpl0 16165 The univariate polynomial ring has the same zero as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
 |-  M  =  ( 1o mPoly  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .0.  =  ( 0g `  P )   =>    |-  .0.  =  ( 0g `  M )
 
Theoremply1mpl1 16166 The univariate polynomial ring has the same one as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
 |-  M  =  ( 1o mPoly  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .1.  =  ( 1r `  P )   =>    |-  .1.  =  ( 1r `  M )
 
Theoremply1ascl 16167 The univariate polynomial ring inherits the multivariate ring's scalar function. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by Fan Zheng, 26-Jun-2016.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   =>    |-  A  =  (algSc `  ( 1o mPoly  R ) )
 
Theoremsubrg1ascl 16168 The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  C  =  (algSc `  U )   =>    |-  ( ph  ->  C  =  ( A  |`  T ) )
 
Theoremsubrg1asclcl 16169 The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   &    |-  B  =  (
 Base `  U )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  (
 ( A `  X )  e.  B  <->  X  e.  T ) )
 
Theoremsubrgvr1 16170 The variables in a subring polynomial algebra are the same as the original ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  X  =  (var1 `  R )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  H  =  ( Rs  T )   =>    |-  ( ph  ->  X  =  (var1 `  H ) )
 
Theoremsubrgvr1cl 16171 The variables in a polynomial algebra are contained in every subring algebra. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  X  =  (var1 `  R )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  H  =  ( Rs  T )   &    |-  U  =  (Poly1 `  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( ph  ->  X  e.  B )
 
Theoremcoe1z 16172 The coefficient vector of 0. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |- 
 .0.  =  ( 0g `  P )   &    |-  Y  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  (coe1 ` 
 .0.  )  =  (
 NN0  X.  { Y } ) )
 
Theoremcoe1add 16173 The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .+b  =  ( +g  `  Y )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .+b  G ) )  =  ( (coe1 `  F )  o F  .+  (coe1 `  G ) ) )
 
Theoremcoe1addfv 16174 A particular coefficient of an addition. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .+b  =  ( +g  `  Y )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .+b  G ) ) `  X )  =  ( (
 (coe1 `
  F ) `  X )  .+  ( (coe1 `  G ) `  X ) ) )
 
Theoremcoe1subfv 16175 A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  B  =  ( Base `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  N  =  ( -g `  R )   =>    |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .-  G ) ) `
  X )  =  ( ( (coe1 `  F ) `  X ) N ( (coe1 `  G ) `  X ) ) )
 
Theoremcoe1mul2lem1 16176 An equivalence for coe1mul2 16178. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  ( ( A  e.  NN0  /\  X  e.  ( NN0  ^m 
 1o ) )  ->  ( X  o R  <_  ( 1o  X.  { A } )  <->  ( X `  (/) )  e.  ( 0
 ... A ) ) )
 
Theoremcoe1mul2lem2 16177* An equivalence for coe1mul2 16178. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  H  =  { d  e.  ( NN0  ^m  1o )  |  d  o R  <_  ( 1o  X.  { k } ) }   =>    |-  (
 k  e.  NN0  ->  ( c  e.  H  |->  ( c `  (/) ) ) : H -1-1-onto-> ( 0 ... k
 ) )
 
Theoremcoe1mul2 16178* The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  S  =  (PwSer1 `  R )   &    |-  .xb  =  ( .r `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  S )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
 gsumg  ( x  e.  (
 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  ( (coe1 `  G ) `  (
 k  -  x ) ) ) ) ) ) )
 
Theoremcoe1mul 16179* The coefficient vector of multiplication in the univariate polynomial ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  Y  =  (Poly1 `  R )   &    |-  .xb  =  ( .r `  Y )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .xb  G ) )  =  ( k  e.  NN0  |->  ( R 
 gsumg  ( x  e.  (
 0 ... k )  |->  ( ( (coe1 `  F ) `  x )  .x.  ( (coe1 `  G ) `  (
 k  -  x ) ) ) ) ) ) )
 
Theoremply1tmcl 16180 Closure of the expression for a univariate polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  ( C  .x.  ( D  .^  X ) )  e.  B )
 
Theoremcoe1tm 16181* Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  ( x  e.  NN0  |->  if ( x  =  D ,  C ,  .0.  )
 ) )
 
Theoremcoe1tmfv1 16182 Nonzero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   =>    |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  D )  =  C )
 
Theoremcoe1tmfv2 16183 Zero coefficient of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  F  e.  NN0 )   &    |-  ( ph  ->  D  =/=  F )   =>    |-  ( ph  ->  ( (coe1 `  ( C  .x.  ( D  .^  X ) ) ) `  F )  =  .0.  )
 
Theoremcoe1tmmul2 16184* Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( A  .xb  ( C 
 .x.  ( D  .^  X ) ) ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  (
 ( (coe1 `  A ) `  ( x  -  D ) )  .X.  C ) ,  .0.  ) ) )
 
Theoremcoe1tmmul 16185* Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( ( C  .x.  ( D  .^  X ) )  .xb  A )
 )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  ( C  .X.  ( (coe1 `  A ) `  ( x  -  D ) ) ) ,  .0.  ) ) )
 
Theoremcoe1tmmul2fv 16186 Function value of a right-multiplication by a term in the shifted domain. (Contributed by Stefan O'Rear, 27-Mar-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  K  =  (
 Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  ( Base `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .X.  =  ( .r `  R )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( A  .xb  ( C  .x.  ( D 
 .^  X ) ) ) ) `  ( D  +  Y )
 )  =  ( ( (coe1 `  A ) `  Y )  .X.  C ) )
 
Theoremcoe1pwmul 16187* Coefficient vector of a polynomial multiplied on the left by a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (coe1 `  ( ( D  .^  X )  .x.  A ) )  =  ( x  e.  NN0  |->  if ( D  <_  x ,  (
 (coe1 `
  A ) `  ( x  -  D ) ) ,  .0.  ) ) )
 
Theoremcoe1pwmulfv 16188 Function value of a right-multiplication by a variable power in the shifted domain. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |- 
 .0.  =  ( 0g `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  P )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  D  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  (
 (coe1 `
  ( ( D 
 .^  X )  .x.  A ) ) `  ( D  +  Y )
 )  =  ( (coe1 `  A ) `  Y ) )
 
Theoremply1scltm 16189 A scalar is a term with zero exponent. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  K ) 
 ->  ( A `  F )  =  ( F  .x.  ( 0  .^  X ) ) )
 
Theoremcoe1sclmul 16190 Coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  Y  e.  B )  ->  (coe1 `  ( ( A `
  X )  .xb  Y ) )  =  ( ( NN0  X.  { X } )  o F  .x.  (coe1 `  Y ) ) )
 
Theoremcoe1sclmulfv 16191 A single coefficient of a polynomial multiplied on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  K  /\  Y  e.  B ) 
 /\  .0.  e.  NN0 )  ->  ( (coe1 `  ( ( A `
  X )  .xb  Y ) ) `  .0.  )  =  ( X  .x.  ( (coe1 `  Y ) `  .0.  ) ) )
 
Theoremcoe1sclmul2 16192 Coefficient vector of a polynomial multiplied on the right by a scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  Y  e.  B )  ->  (coe1 `  ( Y  .xb  ( A `  X ) ) )  =  ( (coe1 `  Y )  o F  .x.  ( NN0  X. 
 { X } )
 ) )
 
Theoremply1sclf 16193 A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  A : K --> B )
 
Theoremcoe1scl 16194* Coefficient vector of a scalar. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K ) 
 ->  (coe1 `  ( A `  X ) )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  X ,  .0.  ) ) )
 
Theoremply1sclid 16195 Recover the base scalar from a scalar polynomial.. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K )  ->  X  =  ( (coe1 `  ( A `  X ) ) `  0 ) )
 
Theoremply1sclf1 16196 The polynomial scalar function is injective. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   =>    |-  ( R  e.  Ring  ->  A : K -1-1-> B )
 
Theoremply1scl0 16197 The zero scalar is zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  Y  =  ( 0g `  P )   =>    |-  ( R  e.  Ring  ->  ( A `  .0.  )  =  Y )
 
Theoremply1scln0 16198 Nonzero scalars create nonzero polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  Y  =  ( 0g `  P )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  K  /\  X  =/=  .0.  )  ->  ( A `  X )  =/=  Y )
 
Theoremply1scl1 16199 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  A  =  (algSc `  P )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( 1r `  P )   =>    |-  ( R  e.  Ring  ->  ( A `  .1.  )  =  N )
 
Theoremply1coe 16200* Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .s `  P )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   &    |-  A  =  (coe1 `  K )   &    |-  R  e.  _V   =>    |-  (
 ( R  e.  CRing  /\  K  e.  B ) 
 ->  K  =  ( P 
 gsumg  ( k  e.  NN0  |->  ( ( A `  k ) 
 .x.  ( k  .^  X ) ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31058
  Copyright terms: Public domain < Previous  Next >